On the stationary gravitational fields

Abstract

The stationary gravitational equations in vacuum are expressed in five different forms. A necessary integral condition on the twist potential φ is derived. The Papapetrou-Ehlers class of stationary solutions is rederived in a different way. In the study of the complex potential theory it is proved from the field equations that a body admitting an arbitrary symmetry must satisfy an integral condition analogous to the equilibrium criterion. It is proved that the vanishing of the scalar curvature of the associated space implies the flatness of the space-time metric. A proof is given for the fact that the only analytic functions of the complex potential F which preserve the field equations form a four-parameter Möbius group. It is also shown that any differentiable function of F and F̄ which preserves the field equations must either be an analytic function of F or the conjugate of such a function. Next the conformastationary vacuum metrics are classified. In the study of the axially symmetric stationary fields a class of metrics (outside the Papapetrou-Ehlers class) is found depending on Euclidean harmonic functions.